Alright, let's address the problem step-by-step for both the rectangular prism and the cylindrical containers.
### Rectangular Prism
1. Given Volume: The volume of the container should be [tex]\(258 \frac{3}{4}\)[/tex] cubic inches. Converting this to a decimal gives us 258.75 cubic inches.
2. Assume Dimensions for Length and Width: For simplicity, we assume:
- Length ([tex]\(l\)[/tex]) = 12 inches
- Width ([tex]\(w\)[/tex]) = 6 inches
3. Calculate the Height:
- The formula for the volume of a rectangular prism is [tex]\( V = l \times w \times h \)[/tex].
- Rearranging for height ([tex]\(h\)[/tex]), we get [tex]\( h = \frac{V}{l \times w} \)[/tex].
- Substituting the values, we get [tex]\( h = \frac{258.75}{12 \times 6} = \frac{258.75}{72} = 3.59375 \)[/tex] inches.
So, for the rectangular prism:
- Length = 12 inches
- Width = 6 inches
- Height = 3.59375 inches
### Cylinder
1. Given Volume: Again, the volume required is 258.75 cubic inches.
2. Assume Radius: For simplicity, we assume the radius ([tex]\(r\)[/tex]) to be 3 inches.
3. Calculate the Height:
- The formula for the volume of a cylinder is [tex]\( V = \pi \times r^2 \times h \)[/tex].
- Rearranging for height ([tex]\(h\)[/tex]), we get [tex]\( h = \frac{V}{\pi \times r^2} \)[/tex].
- Substituting the values, we get [tex]\( h = \frac{258.75}{\pi \times 3^2} = \frac{258.75}{\pi \times 9} \)[/tex].
4. Calculate Numerical Result:
- Simplifying [tex]\( h \)[/tex], we get [tex]\( h = \frac{258.75}{28.274333882308138} \approx 9.151409227783983 \)[/tex] inches.
So, for the cylinder:
- Radius = 3 inches
- Height = 9.151409227783983 inches
### Summary
- Rectangular Prism:
- Length: 12 inches
- Width: 6 inches
- Height: 3.59375 inches
- Volume: 258.75 cubic inches
- Cylinder:
- Radius: 3 inches
- Height: 9.151409227783983 inches
- Volume: 258.75 cubic inches
